Continuity mistake: In the scene where Maurice's Rube Goldberg-ish woodcutting invention works for the first time, we see the piece of wood fly over onto the stack. There is no window there or anything made of glass for that matter, yet in the next shot we hear glass breaking from the wood flying over there.
Continuity mistake: At the end of the Provincial Life song, Gaston is singing from among the crowd and he doesn't have his gun. He climbs up the side of the building without his gun, but when he pops out of the window onto the roof he has his gun. When he pops in front of Belle on the street, the gun is gone again. (00:07:05)
Continuity mistake: In the opening song sequence, there are three blonde girls singing about Gaston next to a water pump. But in a front shot of Gaston walking through the city as they sing, it is a barrel, not a pump, that they are gathered around.
Continuity mistake: After leaving the book store, in an overhead shot Belle passes a shop just as a woman is pouring water towards the downspout, and in the next shot the shop suddenly has a large hanging pipe sign and a table under the window, the green shutters have vanished on the right window, the wood trim has changed, etc.